document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1090652>Question 1090652</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Proof:<BR>\n" );
document.write( "1. ~O > ~ O<BR>\n" );
document.write( "2. X > (X > O)<BR>\n" );
document.write( "/~X </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #705120 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=705120\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "The argument is <font color=red size=4>Invalid</font>\r<BR>\n" );
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document.write( "How can we determine this? By use of a logic table shown below<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/8S6CRrJ.png\"><BR>\n" );
document.write( "Breakdown of the table<BR>\n" );
document.write( "<ul><BR>\n" );
document.write( "<li>Column A represents the truth values for logic expression O (either T for true or F for false)</li><BR>\n" );
document.write( "<li>Column B represents the truth values for logic expression X</li><BR>\n" );
document.write( "<li>Column C is the truth values for logic expression ~O (this is the complete opposite of what is shown in column A)</li><BR>\n" );
document.write( "<li>Column D has us compute the first premise's truth values. In this case, no matter what O happens to be, the truth value of ~O > ~O is always true. This is because ~O > ~O is equivalent to ~~O v ~O and that turns into O v ~O. After substitution of truth values, you'd end up with T v F or F v T which is always true. In plain english, it's like saying \"I either got the object or I didn't get the object\"</li><BR>\n" );
document.write( "<li>Column E isn't a premise so I didn't mark it in blue. It will help set up the premise in column F. Similar to column D, we use the conditional template. The expression X > O is only false if X is true and O is false. This happens in line 4</li><BR>\n" );
document.write( "<li>Column F uses column E. Column F is a premise so I marked it in blue. The same rules used prior (with columns D and E) will be used here</li><BR>\n" );
document.write( "<li>The last column, column G, is the conclusion marked in green. Simply negate or take the opposite of whatever you see in column B (where X is located)</li><BR>\n" );
document.write( "</ul>\r<BR>\n" );
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document.write( "Row 2, where I've marked the premises and conclusion in <font color=red>red</font>, shows us that the argument is invalid. This row is a case where all of the premises are true but the conclusion is false.\r<BR>\n" );
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document.write( "In short,<BR>\n" );
document.write( "if O = true and X = true\r<BR>\n" );
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document.write( "then...<BR>\n" );
document.write( "the premise ~O > ~O is true<BR>\n" );
document.write( "the premise X > (X > O) is true<BR>\n" );
document.write( "but the conclusion ~X is false\r<BR>\n" );
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document.write( "This is why the argument is invalid. It is impossible to do a proof derivation of invalid proofs. It's possible your teacher made a typo or s/he is trying to throw a trick question.\r<BR>\n" );
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document.write( "</font> </SPAN>\n" );
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